# Below is a problem I did. However, it did not match the back of the book. I would like to know where

Below is a problem I did. However, it did not match the back of the book. I would like to know where I went wrong.
Problem:
Solve the following differential equation.
${y}^{\prime }=\frac{y-x}{x}$
$\begin{array}{rl}\frac{dy}{dx}& =\frac{y}{x}-\frac{x}{x}=\frac{y}{x}-1\\ y& =xv\\ \frac{dy}{dx}& =x\frac{dv}{dx}+v\\ x\frac{dv}{dx}+v& =v-1\\ x\frac{dv}{dx}& =-1\\ dv& =-\frac{dx}{x}\\ v& =-\mathrm{ln}x+c\\ \frac{y}{x}& =-\mathrm{ln}|x|+c\\ y& =-x\mathrm{ln}|x|+cx\end{array}$
$y=x\mathrm{ln}|\frac{k}{x}|$
Where did I go wrong?
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candelo6a
You didn't do anything wrong. Your book just took the constant $c$ to be $\mathrm{ln}|k|$.
###### Not exactly what you’re looking for?
Abram Boyd
$y=-x\mathrm{ln}|x|+cx$
$=x\mathrm{ln}|\frac{1}{x}|+cx$
$=x\left(\mathrm{ln}|\frac{1}{x}|+\mathrm{ln}k\right)$, put $c=\mathrm{ln}k$
$y=x\left(\mathrm{ln}|\frac{k}{x}|\right)$, you did everything right